1,073 research outputs found

### Access vs. Bandwidth in Codes for Storage

Maximum distance separable (MDS) codes are widely used in storage systems to
protect against disk (node) failures. A node is said to have capacity $l$ over
some field $\mathbb{F}$, if it can store that amount of symbols of the field.
An $(n,k,l)$ MDS code uses $n$ nodes of capacity $l$ to store $k$ information
nodes. The MDS property guarantees the resiliency to any $n-k$ node failures.
An \emph{optimal bandwidth} (resp. \emph{optimal access}) MDS code communicates
(resp. accesses) the minimum amount of data during the repair process of a
single failed node. It was shown that this amount equals a fraction of
$1/(n-k)$ of data stored in each node. In previous optimal bandwidth
constructions, $l$ scaled polynomially with $k$ in codes with asymptotic rate
$<1$. Moreover, in constructions with a constant number of parities, i.e. rate
approaches 1, $l$ is scaled exponentially w.r.t. $k$. In this paper, we focus
on the later case of constant number of parities $n-k=r$, and ask the following
question: Given the capacity of a node $l$ what is the largest number of
information disks $k$ in an optimal bandwidth (resp. access) $(k+r,k,l)$ MDS
code. We give an upper bound for the general case, and two tight bounds in the
special cases of two important families of codes. Moreover, the bounds show
that in some cases optimal-bandwidth code has larger $k$ than optimal-access
code, and therefore these two measures are not equivalent.Comment: This paper was presented in part at the IEEE International Symposium
on Information Theory (ISIT 2012). submitted to IEEE transactions on
information theor

### MDS Array Codes with Optimal Rebuilding

MDS array codes are widely used in storage systems
to protect data against erasures. We address the rebuilding ratio
problem, namely, in the case of erasures, what is the the fraction
of the remaining information that needs to be accessed in order
to rebuild exactly the lost information? It is clear that when the
number of erasures equals the maximum number of erasures
that an MDS code can correct then the rebuilding ratio is 1
(access all the remaining information). However, the interesting
(and more practical) case is when the number of erasures is
smaller than the erasure correcting capability of the code. For
example, consider an MDS code that can correct two erasures:
What is the smallest amount of information that one needs to
access in order to correct a single erasure? Previous work showed
that the rebuilding ratio is bounded between 1/2 and 3/4 , however,
the exact value was left as an open problem. In this paper, we
solve this open problem and prove that for the case of a single
erasure with a 2-erasure correcting code, the rebuilding ratio is
1/2 . In general, we construct a new family of r-erasure correcting
MDS array codes that has optimal rebuilding ratio of 1/r
in the
case of a single erasure. Our array codes have efficient encoding
and decoding algorithms (for the case r = 2 they use a finite field
of size 3) and an optimal update property

### Optimal Rebuilding of Multiple Erasures in MDS Codes

MDS array codes are widely used in storage systems due to their
computationally efficient encoding and decoding procedures. An MDS code with
$r$ redundancy nodes can correct any $r$ node erasures by accessing all the
remaining information in the surviving nodes. However, in practice, $e$
erasures is a more likely failure event, for $1\le e<r$. Hence, a natural
question is how much information do we need to access in order to rebuild $e$
storage nodes? We define the rebuilding ratio as the fraction of remaining
information accessed during the rebuilding of $e$ erasures. In our previous
work we constructed MDS codes, called zigzag codes, that achieve the optimal
rebuilding ratio of $1/r$ for the rebuilding of any systematic node when $e=1$,
however, all the information needs to be accessed for the rebuilding of the
parity node erasure.
The (normalized) repair bandwidth is defined as the fraction of information
transmitted from the remaining nodes during the rebuilding process. For codes
that are not necessarily MDS, Dimakis et al. proposed the regenerating codes
framework where any $r$ erasures can be corrected by accessing some of the
remaining information, and any $e=1$ erasure can be rebuilt from some subsets
of surviving nodes with optimal repair bandwidth.
In this work, we study 3 questions on rebuilding of codes: (i) We show a
fundamental trade-off between the storage size of the node and the repair
bandwidth similar to the regenerating codes framework, and show that zigzag
codes achieve the optimal rebuilding ratio of $e/r$ for MDS codes, for any
$1\le e\le r$. (ii) We construct systematic codes that achieve optimal
rebuilding ratio of $1/r$, for any systematic or parity node erasure. (iii) We
present error correction algorithms for zigzag codes, and in particular
demonstrate how these codes can be corrected beyond their minimum Hamming
distances.Comment: There is an overlap of this work with our two previous submissions:
Zigzag Codes: MDS Array Codes with Optimal Rebuilding; On Codes for Optimal
Rebuilding Access. arXiv admin note: text overlap with arXiv:1112.037

### On Codes for Optimal Rebuilding Access

MDS (maximum distance separable) array codes
are widely used in storage systems due to their computationally
efficient encoding and decoding procedures. An MDS code with
r redundancy nodes can correct any r erasures by accessing
(reading) all the remaining information in both the systematic
nodes and the parity (redundancy) nodes. However, in practice,
a single erasure is the most likely failure event; hence, a natural
question is how much information do we need to access in order
to rebuild a single storage node? We define the rebuilding ratio
as the fraction of remaining information accessed during the
rebuilding of a single erasure. In our previous work we showed
that the optimal rebuilding ratio of 1/r is achievable (using
our newly constructed array codes) for the rebuilding of any
systematic node, however, all the information needs to be accessed
for the rebuilding of the parity nodes. Namely, constructing array
codes with a rebuilding ratio of 1/r was left as an open problem.
In this paper, we solve this open problem and present array codes
that achieve the lower bound of 1/r for rebuilding any single
systematic or parity node

### Long MDS Codes for Optimal Repair Bandwidth

MDS codes are erasure-correcting codes that can
correct the maximum number of erasures given the number of
redundancy or parity symbols. If an MDS code has r parities
and no more than r erasures occur, then by transmitting all
the remaining data in the code one can recover the original
information. However, it was shown that in order to recover a
single symbol erasure, only a fraction of 1/r of the information
needs to be transmitted. This fraction is called the repair
bandwidth (fraction). Explicit code constructions were given in
previous works. If we view each symbol in the code as a vector
or a column, then the code forms a 2D array and such codes
are especially widely used in storage systems. In this paper, we
ask the following question: given the length of the column l, can
we construct high-rate MDS array codes with optimal repair
bandwidth of 1/r, whose code length is as long as possible? In
this paper, we give code constructions such that the code length
is (r + 1)log_r l

### Explicit MDS Codes for Optimal Repair Bandwidth

MDS codes are erasure-correcting codes that can correct the maximum number of
erasures for a given number of redundancy or parity symbols. If an MDS code has
$r$ parities and no more than $r$ erasures occur, then by transmitting all the
remaining data in the code, the original information can be recovered. However,
it was shown that in order to recover a single symbol erasure, only a fraction
of $1/r$ of the information needs to be transmitted. This fraction is called
the repair bandwidth (fraction). Explicit code constructions were given in
previous works. If we view each symbol in the code as a vector or a column over
some field, then the code forms a 2D array and such codes are especially widely
used in storage systems. In this paper, we address the following question:
given the length of the column $l$, number of parities $r$, can we construct
high-rate MDS array codes with optimal repair bandwidth of $1/r$, whose code
length is as long as possible? In this paper, we give code constructions such
that the code length is $(r+1)\log_r l$.Comment: 17 page

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